An index for Brouwer homeomorphisms and homotopy Brouwer theory

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An index for Brouwer homeomorphisms and homotopy

Brouwer theory

Frédéric Le Roux

To cite this version:

Frédéric Le Roux. An index for Brouwer homeomorphisms and homotopy Brouwer theory. 2015. �hal-00926770v3�

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An index for Brouwer homeomorphisms and

homotopy Brouwer theory

Fr´ed´eric Le Roux

September 23, 2015

Abstract

We use the homotopy Brouwer theory of Handel to define a Poincar´e index between pairs of orbits for an orientation preserving fixed point free homeo-morphism of the plane. Furthermore, we prove that this index is almost additive.

0 Description of the results

0.1 The Poincar´

e index for foliations

Let X be a (smooth) vector field on the plane. If a point x0 is an isolated zero of X,

then one can define the Poincar´e-Hopf index of X at x0 as the winding number of

the vector X(x) when the point x goes once around the singularity x0. This number

is a conjugacy invariant: if Φ is a diffeomorphism, then the Poincar´e-Hopf index of the image vector field Φ∗X at Φ(x0) is equal to the Poincar´e-Hopf index of X at

x0. Analogously, if h is a homeomorphism of the plane, and x0 is an isolated fixed

×x0

Figure 1: Index of a vector field or a homeomorphism along a curve

point of h, the Poincar´e-Lefschetz index of h at x0 is defined as the winding number

of the vector from x to h(x) when the point x goes once around the singularity x0.

Again, for every homeomorphism Φ, the Poincar´e-Lefschetz index of ΦhΦ−1_{at Φ(x} 0)

is equal to the Poincar´e index of h at x0.

Now assume that the planar vector field X has no singularity. Let F be the foliation of the plane by trajectories of X. According to the Poincar´e-Bendixson theory, every leaf of F is properly embedded in the plane. As a consequence, given two distinct leaves F1, F2, one can find an orientation preserving diffeomorphism Φ

that sends F1, F2 respectively to the lines R × {1}, R × {2}. Consider the vector

field Φ∗X. On the lines Φ(F1), Φ(F2) this vector field is horizontal. Thus, given any

curve γ connecting those two lines, the winding number of X along γ is an integer or a half integer. Let us denote this number by I(Φ∗X, γ). An easy connectedness

argument shows that I(Φ∗X, γ) does not depend on the choice of the curve γ. We

claim that this number does not depend either on the choice of the diffeomorphism Φ. This follows from another connectedness argument; the crucial observation is that the space of changes of coordinates, namely orientation preserving diffeomorphisms that globally preserves both lines R × {1}, R × {2}, is connected. In conclusion, the number I(Φ∗X, γ) defines an index associated to the foliation F and the pair

of leaves (F1, F2), and this index is again a conjugacy invariant. This index counts

the algebraic number of “Reeb components” in the area between the two leaves (see Figure 2).

Can we find an analogous definition for homeomorphisms? The discrete counter-part of non vanishing planar vector fields are Brouwer homeomorphisms, i.e. orien-tation preserving fixed point free homeomorphims of the plane. There is a profound

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−1 2 −1 2 1 2 1 2 0 1

Figure 2: Examples of indices for pairs of leaves of a particular planar foliation analogy between the dynamics of Brouwer homeomorphisms and the topology of non singular plane foliations, beginning with the work of Brouwer ([Bro12]). In particular, Brouwer proved that every orbit of a Brouwer homeomorphism h is prop-erly embedded, that is, for every x, the sequences (hn_(x))

n≤0 and (hn(x))n≥0 tends

to infinity1_{. This is of course reminiscent of the Poincar´e-Bendixson theorem. The}

analogy has been developed by several authors, see for example [HT53,LC04,LR05]. Note however that, unlike foliations, the dynamics of Brouwer homeomorphisms is rich enough to prevent any attempt for a complete classification (see [LR01,BLR03]). The aim of the present paper is to define an index for pairs of orbits of Brouwer homeomorphisms that generalizes the Poincar´e-Hopf index for pairs of leaves of a plane foliation. This index is much probably related to the algebraic number of

gen-eralized Reeb components (as defined in [LR05]) separating the two orbits. But it is unclear to the author how to write a precise definition of the index using generalized Reeb components, whereas homotopy Brouwer theory seems to be the perfect tool.

0.2 Homotopy Brouwer theory

In the foliation setting, the key to the definition of the index between pairs of leaves is the existence of a map Φ that straightens both leaves to euclidean lines, and the connectedness of the space of changes of coordinates. Given a Brouwer homeomorphism h and two full orbits O1, O2of h, it is easy to find a homeomorphism

Φ that sends O1 to Z × {1} and O2 to Z × {2}. For a curve γ starting at a point

of Z × {1} and ending at a point of Z × {2}, we denote by I(h′_{, γ) the index of h}′

along γ, that is, the winding number of the vector from x to h′_{(x) when the point x}

runs along the curve γ. Note that h′ _{acts as a horizontal translation on Z × {1} and}

Z_{× {2}, so that the number I(h}′_{, γ) is an integer or a half integer. It is tempting to}

define the index of h between the pair of orbits O1 and O2 as the number I(h′, γ).

However, this time the index I(h′_{, γ) depends on the choice of Φ (and generally also}

1_{We call infinity the point added in the Alexandroff one-point compactification of the plane;}

thus the previous sentence means that every compact subset of the plane contains only finitely many terms of the orbit.

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on the choice of γ, see Figure3). Also note that the corresponding space of changes of coordinates is not connected. Thus one has to give additional conditions on the map Φ to exclude the “bad” maps as the one on Figure 3. These conditions will be provided by Handel’s homotopy Brouwer theory.

• × • × • × • × • × • × • × • × Index 1 Index 0

Figure 3: A bad change of coordinates for the translation T : if Φ is a Dehn twist around the segment [0, 1] × {2}, the index of ΦT Φ−1 _{between (0, 1) and (0, 2) is one,}

whereas the index of T between those same points vanishes

The general setting for homotopy Brouwer theory is the following. Let h be an ori-entation preserving homeomorphism of the plane. Choose distinct orbits O1, . . . Or

of h and assume that they are proper: for every xi ∈ Oi, the sequences (hn(xi))n≤0

and (hn_(x

i))n≥0 tend to infinity (remember that this is automatic if h is a Brouwer

homeomorphism). Let Homeo+(R2_{; O}

1, . . . , Or) denote the group of orientation

pre-serving homeomorphisms g of the plane that globally preserve each Oi, that is,

g(Oi) = Oi for each i. Let Homeo+0(R2; O1, . . . , Or) denote the identity component

of this group: more explicitly, a homeomorphism g belongs to this subgroup if there exists an isotopy (gt)t∈[0,1] such that g0 is the identity and g1 = g, and for every t,

gt fixes every point of O = ∪iOi. We denote the quotient group by

MCG(R2; O1, . . . , Or) = Homeo+(R2; O1, . . . , Or)

.

Homeo+₀(R2; O1, . . . , Or).

The homeomorphism h determines an element of this group which is called the

mapping class of h relative to O and denoted by [h; O1, . . . , Or] (or simply [h]

when the set of orbits is clear from the context). Two elements of the same mapping class will be said to be isotopic relative to O. Two mapping classes [h; O1, . . . , Or] and [h′; O1′, . . . , O′r] are conjugate if there exists an orientation

pre-serving homeomorphism Φ such that Φ(Oi) = O′i for each i, and the mapping classes

[ΦhΦ−1_{; O}′

1, . . . , Or′] and [h′; O′1, . . . , Or′] are equal. The mapping class of a Brouwer

homeomorphism is called a Brouwer mapping class. One of the purposes of ho-motopy Brouwer theory is to give a description of Brouwer mapping classes up to conjugacy. The definition of the index, given in the next subsection, will necessi-tate the classification relative to two orbits, which has been provided by Handel in [Han99]. In subsection0.4 we will state a quasi-additivity property for the index. The classification relative to three orbits will be the key to prove this property, we will describe it in subsection 0.5.

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• × • × • × • × • × • × • × • × T T R R

Figure 4: The orbits Z × {1} and Z × {2}, T and R

0.3 Definition of the index

Let T be the translation by one unit to the right, and R be the “Reeb homeomor-phism” which is partially described on Figure 4. The only relevant features for our purposes are the actions of T and R on the lines R × {1}, R × {2}. We de-note by [T ], [T−1_{], [R], [R}−1_{] the mapping classes relative to the orbits Z × {1} and}

Z_{× {2}. We can now state the classification of Brouwer mapping classes relative to} two orbits2_.

Theorem (Handel, [Han99], Theorem 2.6). Every Brouwer mapping class relative

to two orbits is conjugate to [T ], [T−1_{], [R] or [R}−1_].

• 1 ×1 • 2 ×2 • 3 × 3 • 4 ×4 • 5 ×5 • 1 ×1 • 2 ×2 • 3 × 3 • 4 ×4 • 5 ×5 h

Figure 5: Given two orbits of a Brouwer homeomorphism, Handel’s theorem provides a way to connect the points in each orbit by a homotopy streamline Φ−1_{(R × {1}), Φ}−1_{(R × {2}) which is isotopic to its image (see section} _1.2 _below)

Now let h be a Brouwer homeomorphism, and O1, O2be two orbits of h. Handel’s

theorem provides an orientation preserving homeomorphism Φ that sends O1, O2

2

Note that the context is slightly different from [Han99], because (unlike Handel) we restrict ourseves to conjugacy under orientation preserving homeomorphisms that globally preserve each orbit. The reader may easily deduce the given statement from Handel’s one, or refer to section3.4

below. Furthermore, the mapping classes [T ] and [T−1_{] are actually conjugate, as we will see in}

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respectively onto Z × {1}, Z × {2}, and such that the mapping class of h′ _{= ΦhΦ}−1

relative to these orbits is equal either to [T ], [T−1_{], [R] or [R}−1_{]. Let γ be some}

curve starting at a point of Z × {1} and ending at a point of Z × {2}.

Theorem 1. The number I(ΦhΦ−1_{, γ) does not depend on the choice of the curve}

γ nor of the map Φ.

Theorem 1 will be proved in section 1. We denote this number by I(h, O1, O2)

and call it the Poincar´e index of h between the orbits O1 and O2. As a consequence,

this index is a conjugacy invariant: for every orientation preserving homeomorphism Ψ,

I(ΨhΨ−1, ΨO1, ΨO2) = I(h, O1, O2).

When the homeomorphism h pushes points along a foliation F , it is not difficult to see that this index coincides with the index of the foliation between the leaves containing the orbits O1 and O2. To avoid confusion, we insist that the index is

not an isotopy invariant. In particular it may take any integer or half-integer value,

whereas according to Handel’s theorem there are only finitely many isotopy classes.

0.4 Quasi-additivity

The index of a non vanishing vector field along a curve is additive, in the sense that the index along the concatenation of two curves is the sum of the indices along each of the curves. We will prove that our index satisfies a slightly weaker property. Theorem 2. Let h be a Brouwer homeomorphism, and O1, O2, O3 be three orbits of

h. Then the following quasi-additivity relation holds:

|I(h, O1, O2) + I(h, O2, O3) + I(h, O3, O1)| ≤

1 2.

Note that this bound is optimal, in particular additivity does not hold: an example is provided by the sum of the three leftmost indices on Figure2. Theorem2

will be proved in section 2.

The index for couples of leaves of a foliation satisfies the same property. The proof is much easier, and constitutes an interesting introductory exercise for the proof of Theorem 2.

0.5 Brouwer classes relative to three orbits

The quasi-additivity property essentially comes from the description of the mapping classes of Brouwer homeomorphisms relative to three orbits, which we explain now. A flow is a continuous family (ht)t∈R of homeomorphisms of the plane satisfying the

composition law ht+s = ht◦ hs for every t, s ∈ R. A homeomorphism h is said to

be the time one map of a flow if there exists a flow (ht)t∈R such that h1 = h. A

mapping class is a flow class if it contains the time one map of a flow. If it contains a homeomorphism which is both a time one map and fixed point free, the we will say that it is a fixed point free flow class. The following is an extension of Handel’s theorem.

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Theorem 3. Every Brouwer mapping class relative to one, two or three orbits is a fixed point free flow class.

Theorem 3 will be proved in section 3. We will provide a more explicit formu-lation in section 3.4; the formulation in terms of flow classes is just convenient to express Theorem3in a short way. The number three is optimal: there exist Brouwer homotopy classes relative to four orbits that are not flow classes (see [Han99], ex-ample 2.9 for five orbits and [LR12], exercise 5 for four orbits).

0.6 Aknowledgements

This paper relies heavily on the work of Michael Handel, especially [Han99]. The author is also grateful to Michael Handel for several illuminating conversations, both electronically and during the conference on Topological Methods in Dynamical Systems held in Campinas, Brasil, in 2011. The author also whishes to thank Lucien Guillou for discussions concerning the classification relative to three orbits, and the anonymous referee for his careful reading and his suggestions that have helped clarifying the paper.

1 The index

1.1 Proof of the invariance

Theorem 1 claims that a certain number does not depend on the choice of a curve γ nor of a map Φ. For clarity we state the independence on the curve in a separate lemma. We denote Z × {1} and Z × {2} respectively by Z1 and Z2.

Lemma 1.1. Let h be a Brouwer homeomorphism which globally preserves Z1 and

Z₂ _{and is isotopic, relative to these sets, to one of the four maps T, T}−1_{, R, R}−1_.

Then the index of h along a curve joining a point of Z1 to a point of Z2 does not

depend on the choice of the curve.

Note that Figure 3 above provides a counter-example if we remove from the hypothesis the condition on the isotopy class of h. Given this lemma, the proof of Theorem 1 relies on a proposition concerning conjugacy classes and centralizers in the group MCG(R2_{; Z}

1, Z2). To state the proposition, we introduce the “twist

maps” T1, T2 defined by T1(x, y) = (x + 2 − y, y) and T2(x, y) = (x + y − 1, y): these

maps act like horizontal translations on every horizontal line, T1 translates R × {1}

from one unit to the right and is the identity on R × {2}, and T2 does the converse.

Note that the unit translation T is equal to T1T2, and the map R coincides with

T1T2−1 on the lines R × {1} and R × {2}.

Proposition 1.2. In the group MCG(R2_{; Z}

1, Z2),

• the elements [R] and [R]−1 _{are not conjugate, and not conjugate to [T ];}

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• the centralizer of [R] is the free abelian group generated by [T1] and [T2];

• the centralizer of [T ] coincides with the projection in MCG(R2_{; Z}

1, Z2) of the

centralizer of T in Homeo+(R2_{; Z}

1, Z2): in other words, every mapping class

that commutes with [T ] is the mapping class of a homeomorphism that com-mutes with T .

In the end of this subsection we deduce Theorem1 from Lemma1.1 and Propo-sition 1.2. In subsection 1.2 we will introduce tools from homotopy Brouwer theory which will be used in subsections1.3and1.4to prove the lemma and the proposition. The second point of the proposition, namely that the mapping class of T is conjugate to its inverse, will follow from the stronger fact that T is conjugate to its inverse within the group Homeo+(R2_{; Z}

1, Z2). In the proof of the theorem we will

make use of a homeomorphism Ψ0 ∈ Homeo+(R2; Z1, Z2) realizing the conjugacy.

Note that a half-turn rotation realizes a conjugacy between T and T−1_{, and there is}

such a rotation which preserves Z1∪ Z2, but this rotation exchanges Z1 and Z2, thus

it does not belong to the group Homeo+(R2_{; Z}

1, Z2), and the actual construction of

Ψ0 is less straightforward. Here is one way to do it. Consider the quotient space

R2_{/T , which is an infinite cylinder. Take a homeomorphism of this cylinder that is} isotopic to the identity and that exchanges the projection of the orbits Z1 and Z2.

Lifting this homeomorphism, we get a homeomorphism Ψ1 that commutes with T

and exchanges Z1 and Z2 (more concretely, this map may be obtained as an infinite

composition of “Dehn twists”, pairwise conjugated by powers of T ). Let Ψ2 be

the rotation as above, that exchanges both orbits and satisfy Ψ2T Ψ−12 = T−1. The

composition Ψ0 = Ψ2Ψ1suits our needs, namely we have Ψ0T Ψ−10 = T−1, Ψ0Z1 = Z1

and Ψ0Z2 = Z2. We will also use the specific form of Ψ0 as a composition of a

rotation and a map that commutes with T .

In the following proof we will also make use of the arcwise connectedness of the space of orientation preserving homeomorphisms that commutes with T . This is a consequence of the following facts. Any element of this space induces an element of the space of homeomorphisms of the infinite cylinder R2_{/T that preserve the}

orientation and both ends of the cylinder. This latter space is arcwise connected (see [Ham66]), and every isotopy in the infinite cylinder lifts to an isotopy among

homeomorphisms of the plane that commutes with T .

Proof of Theorem 1. Let h be a Brouwer homeomorphism, and O1, O2 be two orbits

of h. We consider two maps Φ1, Φ2 as in section 0.3 and Theorem 1:

• Φ1, Φ2 are orientation preserving homeomorphisms that send O1, O2

respec-tively to Z1, Z2,

• the maps h1 = Φ1hΦ−11 and h2 = Φ2hΦ−12 are respectively isotopic, relative to

Z₁_{∪ Z}₂_{, to maps U}₁_{, U}₂ _{which belongs to {T, T}−1_{, R, R}−1_}.

According to Lemma 1.1, the index of a Brouwer homeomorphism h′ _in

Homeo+(R2_{; Z}

1, Z2) along a curve joining Z1 to Z2 does not depend on the curve,

thus we may denote it by I(h′_{). We want to show that I(h}

1) = I(h2). The map

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1. Ψ(Z1) = Z1, Ψ(Z2) = Z2,

2. ΨU1Ψ−1 is isotopic to U2 relative to Z1∪ Z2,

and

3. U1, U2 belongs to {T, T−1, R, R−1}.

From now on we work with Ψ, U1, U2 ∈ Homeo+(R2; Z1, Z2) satisfying these

three properties, with a Brouwer homeomorphism h1 that is isotopic to U1

rela-tive to Z1 ∪ Z2, and with h2 = Ψh1Ψ−1 (the reader may forget everything about

h, Φ1, Φ2, O1, O2). We choose some curve γ joining some point in Z1 to some point

in Z2.

We first treat the “Reeb case”, i.e. the case when U1 = R or R−1. According

to the first point of the above Proposition, in this case U1 is not conjugate to any

of the three other maps in {T, T−1_{, R, R}−1_{}. Thus property 2 of the map Ψ implies}

that U2 = U1. Furthermore, the third point of the proposition provides integers

n1, n2 such that Ψ is isotopic to Ψ′ = T1n1T n2

2 relative to Z1 ∪ Z2. We remark that

if (ht) is an isotopy in Homeo+(R2; Z1, Z2), and if every ht is a fixed point free

homeomorphism, then the index I(ht) = I(ht, γ) is defined for every t, and it is an

integer or a half integer that depends continuously on t, thus it is constant. Using this remark, we first see that

I(h2) = I(Ψh1Ψ−1) = I(Ψ′h1Ψ′−1).

We conclude this case by the equality I(Ψ′_h

1Ψ′−1) = I(h1), which follows from the

next lemma since Ψ′ _{commutes with T .}

Lemma 1.3. Let h, Φ ∈ Homeo+(R2_{; Z}

1, Z2) and assume that h is fixed point free

and Φ commutes with T . Then I(ΦhΦ−1_{) = I(h).}

Proof of Lemma 1.3. Since the space of homeomorphisms that commute with T is

arcwise connected, we may choose an isotopy (Φt) from the identity to Φ every

time of which commutes with T . Note that for every point x ∈ Z1∪ Z2 we have

h(x) = Tn_{(x) for some n, and thus (Φ}

thΦ−1t )(Φtx) = Tn(Φtx). We deduce that for

each fixed t the quantity

I(ΦthΦ−1t , Φtγ)

is an integer or a half integer. By continuity this quantity does not depend on t, and we get the lemma.

Now let us turn to the “translation case”, when U1 = T±1. In this case, the

Proposition says that U2 = T±1. Let us first treat the sub-case when U2 = U1. We

have [ΨT Ψ−1_{] = [T ], and the fourth point of the proposition says that the map Ψ}

is isotopic relative to Z1∪ Z2 to a homeomorphism Ψ′ that commutes with T . As

in the Reeb case, we first have the equality I(h2) = I(Ψh1Ψ−1) = I(Ψ′h1Ψ′−1), and

then the equality I(Ψ′_h

1Ψ′−1) = I(h1) follows from Lemma 1.3.

It remains to consider the sub-case when U2 = U1−1, that is, when [ΨT Ψ−1] =

[T−1_{]. Here we will use the homeomorphism Ψ}

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the proof. The mapping class [ΨΨ−1₀ ] commutes with [T ]. According to the proposi-tion, ΨΨ−1₀ is isotopic relative to Z1∪ Z2 to a homeomorphism that commutes with

T , and thus (composing with Ψ0) we see that Ψ is isotopic to a homeomorphism Ψ′

satisfying Ψ′_{T Ψ}′−1_{= T}−1_{. Then we have}

I(h2) = I(Ψh1Ψ−1) = I(Ψ′_h 1Ψ′−1) = I( Ψ′_Ψ−1 0 Ψ0h1Ψ−10 Ψ′Ψ−10 −1 ) = I(Ψ0h1Ψ−10 )

where the last equality follows from Lemma 1.3. It remains to prove the equality I(Ψ0h1Ψ−10 ) = I(h1). Remember that Ψ0was defined as the composition Ψ0 = Ψ2Ψ1,

where Ψ1 commutes with T and Ψ2 is a half-turn rotation. Consider an isotopy from

the identity to Ψ2 among rotations, compose this isotopy on the right with Ψ1 to

get an isotopy from Ψ1 to Ψ2Ψ1 = Ψ0, and concatenate this first isotopy with a

second isotopy from the identity to Ψ1 among homeomorphisms that commute with

T . The resulting isotopy goes from the identity to Ψ0, denote it by (Ψ′t). Then for

every fixed t the map Ψ′ th1Ψ′t

−1

acts like a translation (by a non horizontal vector) on Ψ′

t(Z1 ∪ Z2), so that the index of this map along the curve Ψ′t(γ) is an integer.

We conclude that I(Ψ0h1Ψ−10 , Ψ0(γ)) = I(h1, γ), which completes the proof of the

claim.

1.2 Basic tools for homotopy Brouwer theory

We review some basic objects of homotopy Brouwer theory, namely homotopy

trans-lation arcs and homotopy streamlines. Everything here comes from [Han99].

Let us fix an orientation preserving homeomorphism h, and points x1, . . . xr

whose orbits are properly embedded, i.e. such that lim

n→±∞h n_x

i = ∞.

We denote by O1, . . . , Orthe orbits of x1, . . . , xr, and by O their union. We consider

the set A0 of continuous injective curves α : [0, 1] → R2 which are disjoint from O

except at their end-points which are supposed to be points of O. Two elements α, β of A0 are said to be isotopic relative to O if there exists an isotopy (ht)t∈[0,1] in the

group Homeo+0(R2; O1, . . . , Or), such that h0 = Id and h1(α) = β. Two elements

α, β are homotopically disjoint if α is isotopic relative to O to an element α′ _such

that α′_{∩ β ⊂ O. This is a symmetric relation. A sequence of curves (α}

n)n≥0 in A0

is said to be homotopically proper if for every compact subset K of the plane, there exists n0 such that for every n ≥ n0, there exists α′ ∈ A0 which is isotopic to αn

relative to O and disjoint from K.

Note that the map h acts naturally on A0. An element α of A0 is a

homo-topy translation arc for [h; O1, . . . , Or] if h(α(0)) = α(1) and α is homotopically

disjoint from all its iterates under h. An element α is forward proper if the sequence (hn_(α))

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We insist that these notions (homotopy translation arcs, properness) depend only on the mapping class of h relative to O, rather than on h itself.

For example, take O = Z1∪ Z2. Then the segment [0, 1] × {1} is a homotopy

translation arc for the Reeb homeomorphism R which is both backward and forward proper. It is not difficult to see that every homotopy translation arc for R, relative to Z1 ∪ Z2, and joining the points (0, 1) and (1, 1), is homotopic to this segment

(see [FH03], Lemma 8.7 (2)). The same segment is also a homotopy translation arc for the translation T , but in this latter case there are infinitely many distinct homotopy classes of translation arcs having the same end-points (see Figure 6). These examples will be useful to prove points 1 and 2 of Proposition 1.2.

• ◦ • ◦ • ◦ • ◦ • ◦ • ◦ • ◦ • ◦ T T

Figure 6: Two homotopy translation arcs for [T ] relative to two orbits Still following Handel, we endow the complementary set of O in the plane with a (complete) hyperbolic metric: in other words, we consider a discrete subgroup of isometries G of the hyperbolic plane H2 _{such that the quotient H}2_{/G is}

homeo-morphic to R2 _{\ O. Furthermore, G may be chosen to be of the first kind (see for}

example [Kat92] for the definitions). The main purpose of this geometrisation of R2_{\ O is to provide each isotopy class [α] in A}₀ _{with a unique hyperbolic geodesic} α♯_{. The family of geodesics have wonderful properties. In particular, they have}

minimal intersection so, for instance, two elements of A0 are homotopically disjoint

if and only if the associated hyperbolic geodesics have intersection included in O. The other essential properties are summed up in the following lemma (see [Han99], lemma 3.5; [Mat00], beginning of section 2; [LR12], appendix 1).

Lemma 1.4.

1. Let {αi} be a locally finite family of curves in A0 which are pairwise non

isotopic relative to O, and have pairwise intersections included in O. Then there exists f ∈ Homeo+₀(R2_{; O}

1, . . . , Or) such that for every i, f (αi) = α♯i.

2. More generally, let {αi} be as in the previous item, and let {βj} having the

same properties. Assume that for every i, j, the curve αi is not isotopic to

βj relative to O and in minimal position with βj. Then there exists f ∈

Homeo+₀(R2_{; O}

1, . . . , Or) such that for every i, j, f (αi) = α ♯

i and f (βj) = β ♯ j.

3. Let (αn)n≥0 be a sequence in A0. If (αn)n≥0 is homotopically proper then

(α♯

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Since pairs of geodesics are in minimal position, if α is a homotopy translation arc then (hn_α)♯_{∩ α}♯ _{⊂ O for every n 6= 0. Thus the concatenation}

α♯_{⋆ · · · ⋆ (h}n_α)♯_{⋆ · · ·}

is the image of [0, +∞) under an injective continuous map A+_{. Furthermore, point}

3 of the lemma entails that the homotopy translation arc α is forward proper if and only if the map A+ _{is proper}3_{. The embedding A}+ _{will be called the proper}

forward (geodesic) homotopy streamline generated by α. Proper backward (geodesic) homotopy streamlines are defined analogously. The union of a proper backward

geodesic homotopy streamline and a proper forward geodesic homotopy streamline generated by the same homotopy translation arc will be called a proper (geodesic)

homotopy streamline; it is a topological line, that is, the image of a proper injective

continuous map from the real line to the plane. The word “geodesic” will generally be omitted.

Assume the homotopy translation arc α is both forward and backward proper. Then the set {(hn_α)♯_{, i = 1, . . . , r, n ∈ Z} is a locally finite family of geodesics}

having pairwise intersections included in O (and whose union is the proper homotopy streamline A). The image of this family under h is again a locally finite family of elements of A0 having pairwise intersections included in O. Note furthermore

that for each n and i, h((hn_α)♯_{) is isotopic to the geodesic (h}n+1_α)♯ _{relative to}

O. The first point of the above lemma applies: there exists an element f of the group Homeo+₀(R2_{; O}

1, . . . , Or) which sends h((hnα)♯) to (hn+1α)♯for every n. Then

h′ _{:= f h belongs to the mapping class of h and satisfies h}′n_(α♯_{) = (h}n_α)♯ _{for every}

integer n: in other words, the proper homotopy streamline A is the concatenation of all the iterates of α♯ _{under h}′ _{(the arc α}♯ _{is called a translation arc for h}′ _{in classical}

Brouwer theory, see for example [Gui94]). The construction is easily generalized to give the following corollary.

Corollary 1.5. If A1, . . . , Ar is a family of pairwise disjoint proper homotopy

streamlines for h, then there exists a homeomorphism h′ isotopic to h relative to

O such that h′_(A

i) = Ai for every i = 1, . . . , r.

(This kind of argument is very common in homotopy Brouwer theory; more generally, we will call “straightening principle” every use of the first or the second point of the above lemma).

The beginning of section 3 in [Han99] provides a construction of a hyperbolic structure H in R2_{\ Z for which the translation T is an isometry. Now let O}

1, . . . , Or

be distinct orbits of T . By considering the projections of these orbits in the annulus R2_{/T on the one hand, and the projection of Z in the annulus R}2_/Tr _{on the other}

hand, one can find a map Ψ such that ΨT Ψ−1 _{= T}r _{and Ψ(O}

1 ∪ · · · ∪ Or) = Z.

Then the inverse image of H under Ψ is a hyperbolic structure on the complement of O1∪ · · · ∪ Or for which T is an isometry. Here is one application of this construction.

Let α be a homotopy translation arc for the translation T relative to the orbits

3_{Remember that a map is proper if the inverse image of every compact subset of the target is}

compact; in our context, the map A+ _{is proper if and only if lim}

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O1, . . . , Or. Since T is an isometry, the image of a geodesic is a geodesic, and in

particular we get the relations (Tn_α)♯ _{= T}n_(α♯_{) for every n. Thus the concatenation}

of all the Tn_(α♯_{)’s is a proper geodesic homotopy streamline which is invariant under}

T . In particular we see that every homotopy translation arc for T is homotopic to a translation arc for T . We will take advantage of this remark in the proof of Proposition 1.2.

We end this subsection by a lemma which gives the essential property of flow classes, and which will be useful in section 2.

Lemma 1.6. The Brouwer mapping class [h; O1, . . . , Or] is a fixed point free flow

class if and only if it admits a family of pairwise disjoint proper geodesic homotopy streamlines whose union contains all the Oi’s.

Proof. Let us assume that [h; O1, . . . , Or] is a Brouwer mapping class which admits

a family of pairwise disjoint proper geodesic homotopy streamlines A1, . . . , Ar with

Oi included in Ai for every i. The above corollary provides an element h′ in the

mapping class of h such that h′_(A

i) = Ai for every i. Since h′ coincides with h on

Oi, it has no fixed point on Ai. The fact that h is a flow class now comes from

Lemmas 1.7 and 1.8 below. The first Lemma provides a (fixed point free) time one map of a flow h′′ _{that coincides with h}′ _{on each A}

i. The second Lemma implies that

h′′ _{belongs to the mapping class of h}′_{, and thus also to the mapping class of h.}

Now let us assume that the Brouwer mapping class [h; O1, . . . , Or] is a flow class:

there exists a flow (ht)t∈R such that h1 is isotopic to h relative to the union of the

Oi’s. Consider the trajectory under the flow of a point x ∈ Oi for some i: it is the

image of the real line under the map t 7→ ht(x). Since h1 is isotopic to h relative

to the union of the Oi’s, we have hn(x) = hn(x) for every integer n, and since the

sequence (hn_(x))

n∈Z tends to infinity when n tends to ±∞, this implies that the

trajectory of x under the flow is a topological line.

First assume that the Oi’s belong to distinct trajectories of the flow. Choose for

each i a point xi on Oi, and define a curve αi by letting αi(t) = ht(xi) for t ∈ [0, 1].

Then the αi’s are proper homotopy translation arcs for [h; O1, . . . , Or], and they

generate pairwise disjoint proper homotopy streamlines.

• • • ◦ ◦ × × • • • ◦ ◦ ◦ × ×

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In the bad case when several Oi’s belong to the same trajectory Γ of the

conju-gated flow, one can proceed as follows. The Schoenflies theorem provides a homeo-morphism of the plane that sends Γ to a vertical straight line. Thus, up to a change of coordinates, we may assume that Γ is a vertical straight line. Then Figure 7

indicates how to construct a homotopy translation arc for each of the Oi’s contained

Γ, in such a way that they generate pairwise disjoint homotopy streamlines. Up to isotopy relative to the union of the Oi’s, the whole construction takes place in

an arbitrarily small neighborhood of Γ. This entails that our homotopy translation arcs are backward and forward proper. Thus the homotopy streamlines are proper. Furthermore, the homotopy translation arcs associated to Oi’s included in distinct

trajectories of the flow will still be pairwise disjoint.

Lemma 1.7. Let F be a finite family of pairwise disjoint topological lines in the plane. Let F be the union of the elements of F . Let h′ _{: F → F be a fixed point}

free homeomorphism that preserves each F in F . Then there exists a fixed point free homeomorphism h′′ _{of the plane which extends h}′ _{and which is the time one map of}

a flow.

The proof will use the following construction. Under the hypotheses of the lemma, let U be a connected component of the complementary set of F in the plane. The end compactification of the closure of U is homeomorphic to the unit disk (see the corollary in appendix B): there exist a finite set E in the boundary of the closed unit disk D2_{, and a homeomorphism Φ between D}2_{\ E and the closure of U (each}

interval of the complement of E in the boundary of D2 _{is sent by Φ onto an element}

of F that is part of the boundary of U).

We will also need the following “model flows” on the closed unit disk D2. Let E1, E2, E3 be subsets of the boundary ∂D2 containing respectively one, two and

three points.

• There exists a fixed point free flow on D2_{\ E} 1.

• There exist two fixed point free flows on D2\ E2, such that for the first one the

points on the two components of ∂D2_{\ E}

2 flow in the same direction, whereas

for the second one they flow in opposite directions. • There exist two fixed point free flows on D2 _{\ E}

3, such that for the first one

the points on the three components of ∂D2 _{\ E}

2 flow in the same direction,

whereas for the second one the directions are the same on two components and opposite in the last component.

The flow on D2 _{\ E}

1 is conjugate to a translation flow on a closed half-plane. The

flows on D2_{\ E}

2 are conjugate respectively to the translation flow and to the Reeb

flow on a closed strip. The details of the constructions are left to the reader.

Proof of Lemma 1.7. Let U be any connected component of the complement of F .

Consider the set E ⊂ ∂D2 _{as above. If E contains more than three points, we choose}

some point x0 in E and join x0 to every other point of E by a chord of the disk. The

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disjoint from the elements of the family F . We add these lines to F and extend h′_on

each of the new lines in an arbitrary fixed point free way. We make this construction for every connected component U of R2_{\F , and still denote by F the extended family}

and by h′ _{the extended homeomorphism. Note that every connected component U}

of the complement of (the new) F have at most three boundary components. Since every fixed point free homeomorphism of the real line is conjugate to a translation, we can find a flow (h′′

t), defined on F and preserving each component

of F , such that h′′

1 = h′. It remains to construct an extension of this flow to each

connected component U of the complement of F . We note that the extension can be made independently on each such U.

Now let Φ be a homeomorphism between the closure of U and D2_{\E provided by}

the end-compactification. We note that the cardinal of E coincides with the number of boundary components of U, which equals one, two or three. Up to conjugating by a homeomorphism of the disk that permutes the points of E (and which may reverse the orientation), we can assume that on each component of ∂D2 _{\ E the}

flow (Φh′′

t|∂UΦ−1) flows in the same direction as one of the above “model flows”

(mt). Since all the fixed point free flows of an open interval are conjugate, and

each homeomorphism of ∂D2 _{extends to the disk, a further conjugacy ensures that}

(Φh′′

t|∂UΦ−1) = (mt) on ∂D2. Now we may extend the flow (h′′t) on U by the formula

(Φ−1_m tΦ).

The next Lemma is a variation on Alexander’s trick ([Ale23]).

Lemma 1.8. Let F be a finite family of pairwise disjoint topological lines in the plane. Let h0 be an orientation preserving homeomorphism of the plane. Let Hh0

be the space of orientation preserving homeomorphisms that coincide with h0 on the

union of the elements of F . Then Hh0 is arcwise connected.

Proof. The map h 7→ h−1₀ h is a homeomorphism between Hh0 and HId, thus it suffices

to prove the Lemma when h0is the identity. Denote by F the union of the topological

lines that belong to F , and let U be a connected component of the complementary set of F in the plane. Let Φ be a homeomorphism between D2_{\ E and the closure of}

U, where E is a finite subset of ∂D2 _{as above. If h is an element of H}

Id then Φ−1hΦ

extends to a homeomorphism ¯h of the disk which is the identity on the boundary (in other words, we are using the naturality of Freudenthal’s end compactification of the closure of U). The map h 7→ ¯h is a homeomorphism between the space of homeomorphisms of the closure of U that are the identity on the boundary of U and the space of homeomorphisms of the disk that are the identity on the boundary of the disk. The latter is arcwise connected (and even contractible, see [Ale23]). Thus we may find an isotopy in HId from h to a homeomorphism that is the identity on

U. We proceed independently on each connected component of the complement of F to get an isotopy from h to the identity. (Note that the proof actually shows the the space Hh0 is contractible).

We note that both lemmas still hold, with the same proofs, when the family F is only supposed to be locally finite. We will not make use of this remark.

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1.3 Change of curve

In this subsection we prove Lemma 1.1.

Proof. We assume the hypotheses of the Lemma. The space of continuous curves

in the plane joining two given points is easily seen to be connected. Since the index of h along a curve joining a point of Z1 to a point of Z2 is half an integer, the

usual continuity argument yields that two curves having the same end-points give the same index.

It remains to see that different end-points still give the same index. Let γ0 be

any curve from (0, 1) to (0, 2). Let n0, n1 ∈ Z, and choose some curve α from (n0, 1)

to γ0(0) and some curve β from γ0(1) to (n1, 2). We are left to check that the index

along the concatenation α ⋆ γ0 ⋆ β is equal to the index along γ0. By additivity

of the index, it is enough to prove that the indices along α and β vanish. Let us take care of α (the case of β is obviously symmetric). Since h is isotopic relative to Z₁ _{∪ Z}₂ _{to T, T}−1_{, R or R}−1_{, it is also isotopic to one of these maps relative to Z}₁_. Furthermore R = T on the line R × {1}, thus R and T are isotopic relative to this line (Alexander’s trick, see Lemma 1.8). Finally h is isotopic relative to Z1 to T or

T−1_{. Thus the argument reduces to the following Lemma.}

Lemma 1.9. Let h ∈ Homeo+(R2_{; Z}

1) be a Brouwer homeomorphism whose

map-ping class relative to Z1 is equal to [T ]±1. Then the index of h along any curve

joining two points of the orbit Z1 vanishes.

Proof of Lemma 1.9. We again refer to the additivity to reduce the Lemma to the

case of a curve joining some point (n, 1) to the point (n + 1, 1). The problem is invariant under translation, thus everything boils down to a curve joining (0, 1) to (1, 1). Note that it is enough to do this for some specific curve, since again by

connectedness any two curves joining these two points give the same index.

According to classical Brouwer theory, we can find a translation arc, that is, an injective curve γ joining (0, 1) to (1, 1) and satisfying γ ∩ h(γ) = {(1, 1)} (see [Gui94], Lemme 3.2 or [Han99], Lemma 4.1). Furthermore, every

transla-tion arc is a homotopy translatransla-tion arc (Th´eor`eme 3.3 in the former). According to [Han99], Corollary 6.3, there is a unique isotopy class, relative to Z1, of

homo-topy translation arcs from (0, 1) to (1, 1) (this is also a special case of Lemma 3.7

below). Since h is isotopic relative to Z1 to T or T−1 the segment [0, 1] × {1}

is a homotopy translation arc, and thus γ is isotopic to this segment relative to Z₁_{. The same argument shows that h(γ) is isotopic to the segment [1, 2] × {1}.} We now apply the“straightening principle” (point 1 of Lemma 1.4) to get an iso-topy (Ψt)t∈[0,1] of elements in Homeo+0(R2; Z1) such that Ψ0 is the identity while Ψ1

satisfies Ψ1(γ) = [0, 1] × {1} and Ψ1(h(γ)) = [1, 2] × {1}. By continuity we get

I(h, γ) = I(Ψ1hΨ−11 , [0, 1] × {1}). This last number is equal to zero, since Ψ1hΨ−11

sends the segment [0, 1] × {1} to the segment [1, 2] × {1}. This completes the proof of the Lemma.

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1.4 Centralizers of

[T ] and [R]

Proof of proposition 1.2. The translation T admits infinitely many isotopy classes

of homotopy translation arcs sharing the same endpoints, while R admits only one (see section 1.2). This property distinguishes the conjugacy classes of [T ] and [R] in

the mapping class group.

To see that [R] and [R]−1 _{are not conjugate we will argue by contradiction. The}

proof is a “homotopic version” of the easier fact that there is no conjugacy between R and R−1 by a homeomorphism Ψ that preserves orientation and globally fixes each line R × {1} and R × {2}, which may be proved by contradiction as follows: such a homeomorphism would reverse the orientation on the line R × {1}, and since it preserves the orientation of the plane it would have to exchange the two half-planes delimited by this line, in contradiction with the preservation of R × {2}.

Let α1 = [0, 1] × {1}, α2 = [0, 1] × {2}. These segments are homotopy translation

arcs for R. Let A1 = R × {1}, A2 = R × {2} be the proper homotopy streamlines

generated by α1, α2. These lines are oriented by the action of R, and each one is

on the left hand side of the other one with respect to this orientation. Assume by contradiction that the relation [Ψ][R][Ψ]−1 _{= [R]}−1 _{holds in the mapping class}

group, with Ψ ∈ Homeo+(R2_{; Z}

1, Z2). Then Ψ(α1) is a homotopy translation arc for

R−1_{. Since Ψ globally preserves Z}

1, the end-points of this arc are on Z1. Remember

that there is a unique homotopy class of homotopy translation arcs for R or R−1_,

relative to Z1∪ Z2, joining a given point of Z1 to its image ([FH03], Lemma 8.7 (2)).

Thus there exists some n0 such that Ψ(α1) is isotopic to Rn0(α1) = [n0, n0+ 1] × {1}

with the opposite orientation. Up to composing Ψ with T−n0 (which commutes with

R), we may assume that n0 = 0. Then from the relation [Ψ][R][Ψ]−1 = [R]−1 we

get that for every integer n, the arc Ψ(Rn_α

1) is isotopic to the arc R−nα1 endowed

with the opposite orientation. Up to composing Ψ by a homeomorphism provided by the “straightening principle” (point 1 of lemma 1.4), we may further assume that Ψ(Rn_α

1) = R−nα1 for every n. Thus Ψ globally preserves the line A1 while

reversing its orientation. Since Ψ preserves the orientation of the plane, it must send the half-plane on the left hand side of A1 to its right hand side. But this contradicts

the fact that Ψ globally preserves Z2. The proof of the first point of Proposition 1.2

is complete.

The argument for the second point, namely that the mapping class of T is con-jugate to its inverse, has already been given (see the construction of Ψ0 after the

statement of the Proposition). We turn to the third point of the Proposition: we will prove that the centralizer of [R] in the mapping class group is generated by [T1] and [T2]. The homeomorphism T1RT1−1 coincides with R on the union of the

lines R × {1} and R × {2}. Alexander’s trick (Lemma 1.8) provides an isotopy from T1RT1−1 to R relative to R × {1, 2}, and in particular this is an isotopy relative

to Z1 ∪ Z2. In other words, the mapping class [T1] commutes with [R]. The same

argument shows that [T2] commutes with [R]. It is easy to check that the subgroup

generated by those two elements is free abelian. Conversely, let Ψ ∈ Homeo+(R2_{; Z}

1, Z2) such that ΨRΨ−1 is isotopic to R, and

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unique-ness of the homotopy translation arcs, there exists n1, n2 such that, for every integer

n, the arc Ψ(Rn_α

1) is isotopic to Rn+n1α1, and Ψ(Rnα2) is isotopic to Rn+n2α2. The

“straightening principle” (point 1 of lemma 1.4) provides a first isotopy relative to Z₁ _{∪ Z}₂ _{from Ψ to a homeomorphism that coincides with R}n1 on A

1 and with Rn2

on A2. Since T1n1T n2

2 also coincides with Rn1 on A1 and with Rn2 on A2,

Alexan-der’s trick (Lemma 1.8) provides a second isotopy to Tn1

1 T2n2, which proves that

[Ψ] = [Tn1

1 T n2

2 ], as wanted.

Finally, let us suppose that T′ _{= ΨT Ψ}−1 _{is isotopic to T relative to Z}

1∪ Z2, and

let us find a homeomorphism that is isotopic to Ψ and that commutes with T . Let α′

1 = Ψ(α1), α′2 = Ψ(α2), and consider the family of curves

{T′n(α′₁), T′n(α′₂) | n ∈ Z}.

By conjugacy, this is a locally finite family of pairwise disjoint curves. The “straight-ening principle” (point 1 of lemma 1.4) provides a homeomorphism Ψ1, isotopic to

the identity relative to Z1 ∪ Z2, sending each curve in our family to its isotopic

geodesic. On the other hand, since T′ _{is isotopic to T , the curve T}′n_(α′

1) is isotopic

to Tn_(α′

1). We may have chosen the geodesic structure so that T is an isometry (see

section 1.2), and then the corresponding geodesic is (Tn_(α′

1))♯= Tn(α1′ ♯

). Likewise, the geodesic in the isotopy class of T′n_(α′

2) is Tn(α′2 ♯

). Letting Ψ′ _{= Ψ}

1Ψ we get,

for every integer n, Ψ′_(Tn_(α

1)) = Tn(α′1 ♯

) and Ψ′_(Tn_(α

2)) = Tn(α2′ ♯

). Thus the con-catenation of the curves Tn_(α′

1 ♯

) is a proper homotopy streamline which is invariant under T , the same holds for the concatenation of the curves Tn_(α′

2 ♯

), both homotopy streamlines are disjoint, and the first one is on the right hand side of the second one endowed with the orientation induced by the action of T . Consider the quotient map P : R2 → R2/T . The projections of α′1

♯

and α′2 ♯

in this quotient are two disjoint simple closed curves, and P α′

1 ♯

is on the right hand side of P α′ 2 ♯ , where P α′ 2 ♯ is oriented from Ψ(0, 2) to Ψ(1, 2). Of course, the projections of the curves α1 and α2

share the same properties. The classification of surfaces provides a homeomorphism of the quotient annulus that is isotopic to the identity and sends P α1 on P α′1

♯

and P α2 on P α′2

♯

; we may further demand that the image of the projection of a point p ∈ α1 is precisely the projection of the point Ψ′(p), and likewise for α2. Lifting

this homeomorphism to the plane, we get a homeomorphism Ψ′′ _{which commutes}

with T and agrees with Ψ′ _{on the lines A}

1 and A2. A last use of Alexander’s tricks

provides an isotopy (relative to A1∪ A2) from Ψ′ to Ψ′′. Finally [Ψ′′] = [Ψ′] = [Ψ]

and Ψ′′ _{commutes with T , which completes the proof.}

2 Quasi-additivity

In this section we deduce the quasi-additivity Theorem 2 from the classification in homotopy Brouwer theory provided by Theorem 3. We will use the characterization of flow classes in terms of proper homotopy streamlines (Lemma 1.6).

Let h be a Brouwer homeomorphism and let us choose two orbits O1, O2 of h.

As-sume there exist two disjoint proper homotopy streamlines Γ1, Γ2 for [h; O1, O2]

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(Corol-lary1.5), there exists h′ _{∈ [h; O}

1, O2] which globally fixes Γ1and Γ2. The

Schoenflies-Homma theorem (appendix B) provides an orientation preserving homeomorphism Φ that sends O1 onto Z1, O2 onto Z2, Γ1 onto R × {1} and Γ2 onto R × {2}; we may

further demand that Φh′_Φ−1 _{coincides either with T or T}−1_{on each line R×{1} and}

R_{× {2}. Then, by Alexander’s trick (Lemma} _1.8_{), Φh}′_Φ−1 _{is isotopic to T, T}−1_{, R} or R−1 _{relative to Z}

1∪ Z2, and so is ΦhΦ−1. Thus the index I(h, O1, O2) is defined

as the index of ΦhΦ−1 _{along any curve going from a point of Φ(O}

1) to a point of

Φ(O2) (section 0.3).

We would like to be able to evaluate this index directly with h, Γ1 and Γ2. For

this, we consider some curve α going from a point of O1 to a point of O2, and an

isotopy (Φt) from the identity to Φ. The number

I(ΦthΦ−1t , Φt(α))

varies continuously from I(h, α) to I(ΦhΦ−1_{, Φ(α)) = I(h, O}

1, O2). Furthermore,

the total variation I(h, O1, O2) − I(h, α) of this number may be expressed as follows.

Let as denote the angular variation, when t goes from 0 to 1, of the vector joining

the points Φt(α(s)) and ΦthΦ−1t (Φt(α(s))). Then

I(h, O1, O2) − I(h, α) = a1− a0.

Indeed, consider the map

∆ : [0, 1]2 _{→ R}2_{\ {0}}

(s, t) 7→ ΦthΦ−1t (Φt(α(s))) − Φt(α(s)).

Since the unit square is contractible, the winding number of ∆ along the boundary of the unit square is zero. On the other hand this number is the difference between the two terms of the above equality.

As an example, consider the situation depicted by Figure8. One can find a map Φ that sends the curves Γ1, Γ2 to parallel straight lines, and an isotopy from the

identity to Φ during which the vector joining α1(0) to ΦthΦ−1t (α1(0)) is constant,

while the vector joining α1(1) to ΦthΦ−1t (α1(1)) undergoes an angular variation of

a sixth of a full turn in the positive direction. Thus I(h, O1, O2) = I(h, α1) + 1/6.

Now consider three orbits O1, O2, O3 of h. According to the classification of

Theorem 3, and its practical version given by Lemma 1.6, we may find pairwise disjoint proper homotopy streamlines Γ1, Γ2, Γ3 for [h; O1, O2, O3] containing

respec-tively O1, O2, O3. The topological lines Γ1, Γ2 are also proper homotopy streamlines

with respect to [h; O1, O2], and thus the above discussion applies. Now there are two

cases. The first case happens when one of the three lines separates4 _{the other two,}

and the situation is topologically equivalent to three parallel straight lines. Since our index is a conjugacy invariant, we may assume that the streamlines are parallel

4_{A subset A of a topological space X is said to separate two other subsets B, C if A is disjoint}

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• • • • • • _• • • Γ1 × × × × × × × × × Γ2 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Γ3 α1 α2

Figure 8: Three streamlines in the non-separating case

straight lines. In this situation we choose any curve α1 from O1 to O2, and any

curve α2 from O2 to O3, with α1(1) = α2(0). We have

I(h, O1, O2) = I(h, α1), I(h, O2, O3) = I(h, α2),

I(h, O1, O3) = I(h, α1⋆ α2) = I(h, α1) + I(h, α2).

Since I(h, O3, O1) = −I(h, O1, O3), in this case we get perfect additivity, namely

I(h, O1, O2) + I(h, O2, O3) + I(h, O3, O1) = 0.

In the second case, none of the three lines separates the other two, as on figure8. There exists a homeomorphism of the plane that sends the general situation to the particular situation depicted on the Figure. To begin with let us assume that this homeomorphism preserves the orientation of the plane. Then we may again assume the three lines are those depicted on the Figure. We pick two curves α1, α2 as above.

According to the above discussion, this time we get

I(h, O1, O2) = I(h, α1) + 1/6, I(h, O2, O3) = I(h, α2) +

1 6, I(h, O3, O1) = −I(h, α1⋆ α2) + 1 6 and thus

I(h, O1, O2) + I(h, O2, O3) + I(h, O3, O1) =

1 2.

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It remains to consider the case when the situation of the Figure happens only up to an orientation reversing homeomorphism. This case yields an entirely similar computation, where the ’+1/6’ are replaced by ’−1/6’, and thus the final ’+1/2’ becomes a ’−1/2’. In any case the quasi-additivity relation holds.

3 Brouwer classes relative to three orbits

In this section we prove Theorem 3.

3.1 Advanced tools for homotopy Brouwer theory

We need to go deeper into homotopy Brouwer theory, borrowing from [Han99] and [FH12]. The statements that are suitable for our needs are not explicitly stated in the quoted papers. In Appendix A we will explain how to get these statements (Proposition 3.1 and 3.3 below) from those papers.

We adopt the notations of section1.2, and we generalize the notion of homotopy translation arc to encompass the case of an arc meeting several Oi’s. Consider an

arc α which is the concatenation α1⋆· · ·⋆αkof some elements αiin A0. Assume that

α(1) = h(α(0)), and that for every n, i, j, the arcs hn_(α

i) and αj are homotopically

disjoint. Then α is called a (generalized) homotopy translation arc for [h; O1, . . . , Or].

A typical example is shown on figure9.

• ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ • α1 α2

Figure 9: A generalized homotopy translation arc containing two orbits The (generalized) homotopy translation arc will be called forward proper if α1, · · · , αk are forward proper. Equivalently, the half-infinite concatenation

A+= α₁♯⋆ · · · ⋆ α♯_k⋆ (hα1)♯⋆ · · · ⋆ (hαk)♯⋆ · · ·

is the image of [0, +∞) under a proper injective continuous map. This set will be called a (generalized) proper forward homotopy streamline. A (generalized) proper

backward homotopy streamline is defined symmetrically. We will need the existence

of a nice family of (generalized) proper backward and forward homotopy streamlines. Proposition 3.1 (Handel). Let h be a Brouwer homeomorphism, and x1, . . . , xr

points belonging to distinct orbits O1, . . . , Or of h. Then there exist a number 1 ≤

r′ _{≤ r and a family of 2r}′ _{pairwise disjoint (generalized) proper backward or forward}

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union of the Oi’s. Furthermore, the backward and forward streamlines alternate in

the cyclic order at infinity.

The backward homotopy streamlines provided by this Proposition will be de-noted by A−₁, . . . , A−_r′, and the forward homotopy streamlines by A

+

1, . . . , A+r′. The

condition on the cyclic order at infinity means that we may choose the numbering so that there exists an arbitrarily small neighbourhood of ∞, whose boundary is an (oriented) Jordan curve that meets each streamline exactly once, and that meets A+₁, A−₁, A+₂, A−₂, . . . , A−_r′ in that order. A typical example is displayed on figure 10.

• × • × • × • × • × • × • × • × • × ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ A−₁ A+₂ A+₁ A−₂

Figure 10: Alternating forward and backward proper homotopy streamlines for a Brouwer class: here r = 3 and r′ _{= 2}

A reducing line for [h; O1, . . . , Or] is a topological line, i. e. the image of a proper

continuous injective map ∆ : R → R2_{, which is disjoint from O = O}

1∪· · ·∪Or, such

that none of the two topological half-planes delimited by ∆ contains all the Oi’s,

and such that ∆ is properly isotopic to h(∆) relative to O: there exists a continuous proper map H : R × [0, 1] → R2 _{such that H(., 0) = ∆, H(., 1) = h∆, and for every}

t the map H(., t) is injective.

Let A1 denote the set of topological lines. Exactly as for A0, each element in

A1 is properly isotopic relative to O to a unique geodesic for the hyperbolic metric,

and the “straightening principle” still holds (see [Han99] for details). Lemma 3.2. Lemma 1.4 holds when A0 is replaced by A0∪ A1.

Thus if ∆ is a reducing line we may find an element h′ _{of the mapping class}

[h; O1, . . . , Or] such that h′(∆) = ∆. In particular each orbit Oi has to be entirely

on one side of ∆.

We will say that [h; O1, . . . , Or] is a translation class if it contains a

homeo-morphism which is conjugate to a translation. (Equivalently, [h; O1, . . . , Or] is a

translation class if and only if it is conjugate to [T ; Z × {1}, . . . , Z × {r}].)

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1. If r′ _{= 1, then [h; O}

1, . . . , Or] is a translation class.

2. If r′ _{= r, then there exists a reducing line which is disjoint from the 2r}

homo-topy backward and forward streamlines of the preceding proposition.

3. More generally, as soon as r ≥ 2, there exists a reducing line which is disjoint from the r′ _{backward proper homotopy streamlines, and from all the forward}

proper homotopy streamlines that meet only one of the Oi’s.

We call the first case the translation case, the second case the alternating case, and the remaining case the remaining case. Note that in the alternating case each ho-motopy translation arc provided by the first above proposition meets only one orbit; in other words, these are homotopy translation arcs in the non generalized mean-ing. The alternating case generalizes the multiple Reeb class case (see Figure 11) in which the cyclic order at infinity on the set of proper homotopy streamlines given by Proposition 3.1 is A−1, A+2, . . . , Aεr, Ar−ε, . . . , A−2, A+1, where A−i and A+i are the

back-ward and forback-ward homotopy streamlines meeting the same orbit Oi, and ε = ±1

according to the parity of r. The multiple Reeb class case is treated by Franks and Handel (Lemma 8.9 in [FH03]). A−₁ A+₁ A−2 A+2 A−₃ A+₃ A−4 A+4 A−₁ A+₁ A−2 A+2 A−₃ A+ 3 A− 4 A+₄

Figure 11: A multiple Reeb class (left) and another alternating case (right) Lemma1.6 above provides a caracterization of flow classes in terms of homotopy streamlines. The proof may be easily adapted to the case of generalized homotopy streamlines, and we leave this task to the reader. For future reference we state the corresponding result.

Lemma 3.4. Lemma 1.6 still holds when we replace homotopy streamlines with

generalized homotopy streamlines.

The easiest way to construct a reducing line is to consider a proper homotopy streamline and to “push it on the side”, as explicited by the following lemma. Lemma 3.5. Let A be a proper homotopy streamline for some [h; O1, . . . , Or]. Let

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some points of O. Then there exists a reducing line ∆ that separates O′ _{:= U} 1∩ O

from O \ O′_{. Furthermore ∆ may be chosen to be included in any given open set}

containing A.

Proof. By the straightening principle (Corollary 1.5), there is some h′ _∈

[h; O1, . . . , Or] such that A is invariant under h′. By applying a conjugacy we may

assume that A is a straight vertical line on which the restriction of h′ is a translation. A further conjugacy ensures the existence of a vertical strip B containing A such that B ∩ O = A ∩ O. A variation on Alexander’s trick provides some h′′ _{∈ [h; O}

1, . . . , Or]

such that h′′ _{= h}′ _{outside B and h}′′ _{is a translation in some smaller vertical strip}

B′ _{still containing A. Now it is clear that one of the two boundary components of}

B′ _{is a reducing line for [h}′′_{; O}

1, . . . , Or] = [h; O1, . . . , Or]. Given an open set V

containing A one can push ∆ inside V by an isotopy relative to O, which proves the last sentence of the lemma.

We will now turn to the proof of Theorem3. In the translation case the mapping class of h contains a translation, which is certainly the time one map of a flow, so there is nothing to prove in this case. We will give a separate argument in the alternating case and in the remaining case. According to Lemma 1.6, it suffices to find a family of pairwise disjoint (generalized) proper homotopy streamlines. For this we will consider the (generalized) homotopy translation arcs α−₁, . . . , α−_r′ that

generate the backward proper homotopy streamlines A−₁, . . . , A−_r′. In the alternating

case, we will prove that the αj’s are also forward proper, thus they will generate the

wanted family of homotopy streamlines. Life will be a little more complicated in the remaining case, especially in one sub-case where one homotopy streamline will be provided by forward iterating one of the A−_j’s, another one by backward iterating one of the A+j’s, and the last one by a separate argument.

3.2 The alternating case

In this case we prove a stronger result, namely that the mapping class of h is a flow class, not only when r = 3 as is required by the Theorem, but for every r ≥ 1. Note that in the present section all homotopy translation arcs will be non generalized ones. We choose once and for all a Brouwer homeomorphism h, and we introduce the following definitions. A family O1, . . . , Or is alternating if there

exists a family of 2r pairwise disjoint (non generalized) proper backward and forward homotopy streamlines as in the conclusion of Proposition 3.1. This family satisfies

the uniqueness of homotopy translation arcs if for every point x ∈ O1∪· · ·∪Or there

exists a unique homotopy class, relative to these orbits, of homotopy translation arcs joining x and h(x).

Lemma 3.6. Every family of alternating orbits satisfies the uniqueness of homotopy translation arcs.

Given the lemma, we consider a family of alternating orbits, and 2r backward and forward homotopy streamlines generated by homotopy translation arcs α±i . By

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to α+_i . In particular, α_i− is both backward and forward proper. Furthermore, the proper streamlines generated by the α−i ’s are pairwise disjoint, since they arise

from the iteration of pairwise disjoint backward homotopy streamlines. We apply Lemma1.6to conclude that [h; O1, . . . , Or] is a flow class, which completes the proof

that the mapping class of h relative to any alternating family of orbits is a flow class. Now let us prove the Lemma. The proof is by induction on the number r of orbits. For r = 1 we have a translation class, and the uniqueness of homotopy translation arcs is Corollary 6.3 of [Han99] (see also Corollary 2.1 in [LR12]). Let r ≥ 2 and let O1, . . . , Or be a family of r alternating orbits. We consider the 2r pairwise disjoint

proper backward and forward homotopy streamlines given by the definition. Let ∆ be a reducing line provided by point 2 of Proposition 3.3. By the “straightening principle” (Lemma 3.2) we may find some h′ _{in the mapping class [h; O}

1, . . . , Or]

such that h′_{(∆) = ∆. The topological line ∆ splits the set O = O}

1∪· · ·∪Orinto two

subsets, say O = O′ _{⊔ O}′′_{, each one consisting in less than r orbits of h. Note that}

h and h′ still have the same mapping class relative to O′. Furthermore, since the reducing line ∆ is disjoint from the backward and forward homotopy streamlines, it also splits the family S of streamlines into two sub-families that we denote by S′ _and

S′′_{. The key point is that S}′ _{is an interval of S in the cyclic order at infinity, and}

since the backward and forward streamlines in S alternate, this is still true in S′_{. In}

particular, the family of orbits making up O′ _{is alternating. Choose some orbit O} i

in O′ _{and denote by α}−_{, α}+ _{the backward and forward proper homotopy translation}

arcs inducing the elements of S′ _{that meet O}

i. By the induction hypothesis, the

family of orbits in O′ _{satisfies the uniqueness of homotopy translation arcs, and thus}

there is an iterate hN_α− _{which is homotopic to α}+ _{with respect to O}′_{. Obviously}

h′N_α− _{is also homotopic to α}+ _{relative to O}′_{. Since both arcs are disjoint from}

∆, this homotopy may be chosen to be disjoint from ∆ (compose the homotopy with a retraction from the plane to the half plane delimited by ∆ and containing both arcs). Then it is a homotopy relative to O. Thus hN_α− _{is homotopic to α}+

relative to O. In particular, the backward proper homotopy translation arc α−

is also forward proper. The same argument applies to S′′_{. Finally the r proper}

backward homotopy translation arcs provided by Proposition 3.1 are also forward proper. Since the corresponding proper backward homotopy streamlines are pairwise disjoint, the proper homotopy streamlines are also pairwise disjoint. Note that the argument also shows that the union of the proper homotopy streamlines contain all the A−i ’s and the A+i ’s. We are now in a position to apply the following lemma,

which shows that O1, . . . , Or satisfies the uniqueness of homotopy translation arcs.

This concludes the induction.

Lemma 3.7. Let O1, . . . , Or be distinct orbits of h. Assume that there exists a

proper homotopy translation arc αi for each orbit Oi, such that the associated proper

homotopy streamlines A1, . . . Ar are pairwise disjoint. Also assume that the positive

and negative ends of these streamlines alternates in the cyclic order at infinity. Then every (non generalized) homotopy translation arc α for [h; O1, . . . , Or] is isotopic